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**princess snowwhite****Member**- Registered: 2012-11-06
- Posts: 29

Can anyone tell me how to solve a problem using c program by "eigen pair by power method"?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,493

Hi princess snowwhite;

I can not show you with A C++ program. The program would do everything for you in a flash, so what is there to see.

But we can discuss it here and do it manually ( within reason ) and then you will see everything.

First, the power method is for finding the largest eigenvalue is it not? So what is this eigenpair?

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,522

Mabe she means eigenvalue and the corresponding eigenvector? :

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,493

Hi;

Maybe but the power method the way I remember it can only get the dominant or largest eigenvalue.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,522

Well, from the eigenvalue you can get the corresponding eigenvector...

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,493

We won't have the eigenvalue, the power method gets through iteration an approximation to either the largest or the smallest eigenvalue.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**princess snowwhite****Member**- Registered: 2012-11-06
- Posts: 29

I don't know what is this eigen pair, THAT'S WHY i am asking you

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,522

Where'd you get the term "eigen pair" from, then?

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,493

Hi princess snowwhite;

I have some notes on computing an eigenpair. But do you understand the power method? Do you need to see that first.

Also when I was in the business of computing eigenvalues I did not bother with terminology or jargon as I call it. When I saw the word "eigenpair" I just scoffed and ignored it. It does seem that the power method gets both the dominant eigenvalue and the dominant eigenpair.

anonimnystefy wrote:

Mabe she means eigenvalue and the corresponding eigenvector?

It appears that Mr anonimnystefy is correct, every eigenvector is associated with an eigenvalue. These are called an eigenpair.

So now the question is what do you want. An example of the power method producing an eigenpair? Just the definition provided above?

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**princess snowwhite****Member**- Registered: 2012-11-06
- Posts: 29

I needed the definition and example also

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,493

Okay, I have a nice example please hold on while I post it.

Let's start with this simple book example and we will use a very simplified power method to get the dominant eigenpair. First we go after the dominant eigenvector using the power method.

We choose the initial vector with a guess of

We begin to iterate using x0 and A to generate x1.

Now for the purposes of keeping the elements of xn small we will divide x1 by the bottom entry in this case a 5. This was an arbitrary choice and in a real calculation we would use a better system.

Now we continue to iterate.

We divide x2 by the bottom element in this case 5.8

Continuing the above steps we get:

It really does look like the dominant eigenvector is approaching

From that eigenvector we get the dominant eigenvalue by applying this formula:

where x is the dominant eigenvector and A is original matrix ( above ).

That is the dominant eigenvalue. We are done.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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